Scalable dynamical systems for multi-agent steering and simulation

Scalable Dynamical Systems for Multi-Agent Steering and Simulation Siome Goldenstein Menelaos Karavelas Dimitris Metaxas Leonidas Guibas

Ambarish Goswami

University of Pennsylvania siome,dnm@graphics.cis.upenn,

Stanford University menelaos,guibas@,

Discreet ambarish.goswami@

Abstract

We present a new methodology for agent modeling that is scalable and efficient.It is based on the integration of nonlinear dynam-ical systems and kinetic data structures.The method consists of three-layers that model steering,flocking,and crowding agent be-haviors among moving and static obstacles in2and3D.Thefirst layer,the local layer is based on the use of nonlinear dynami-cal systems theory and models low level behaviors,it is fast and efficient,and does not depend on the total number of agents in the environment.The use of dynamical systems allows the use of continuous numerical parameters with which we can modify the interaction of each agent with the environment.This creates controllable distinctive behaviors.The second layer,a global en-vironment layer consists of a specifically designed kinetic data structure to track efficiently the immediate environment of each agent and know which obstacles/agents are near or visible to the given agent.This layer reduces the complexity in the local layer. In the third layer,a global planning layer,the problem of target tracking is generalized in a way that allows navigation in maze-like terrains,avoidance of local minima and cooperation between agents.We implement this layer based on two approaches that are suitable for different applications.One is to track the closest single moving or static target.The second is to use a pre-specified vectorfield.This vector can be generated automatically(with har-monic functions,for example)or based on user input to achieve tht desired output.

We demonstrate the power of the approach through a series of experiments simulating single/multiple agents and crowds mov-ing towards moving/static targets in complex environments.

1Introduction

The modeling of autonomous digital agents and the simulation of their behavior in virtual environments is becoming increas-ingly important in computer graphics and robotics.In virtual re-ality applications,for example,each agent interacts with the other agents and the plex interactions in real time are necessary to achieve nontrivial behavioral scenarios.Mod-ern game applications require the creation of smart autonomous agents with varying degrees of intelligence to allow for multiple levels of game complexity and a variety of behaviors.These be-haviors must allow for complex interactions and must be adaptive in terms of both time and space(continuous changes in the envi-ronment).Finally,the modeling approach should scale well with the complexity of the environment geometry,the number and in-telligence of the agents,and the level of the various environment-agent interactions.

There have been several promising approaches towards achiev-ing the above goal,but many of them are restrictive in terms of their application domain and they do not scale well with the complexity of the environment.This paper attempts to develop a mathematically rigorous approach to modeling complex low-level behaviors in real time that is scalable,adaptive and suitable to a distributed application.

We develop a three-layer approach to modeling autonomous agents based on the integration of nonlinear dynamical system theory,kinetic data structures,and harmonic functions.Thefirst layer consists of differential equations based on nonlinear dy-namic system theory which model the behavior of the autonomous agent in a complex environment.The second layer incorporates the motion of the agents,the obstacles and the targets into a ki-netic data structure,and provides a very efficient and scalable ap-proach for adapting an agent’s motion based on its changing local environment.Finally,differential equations based on harmonic functions,the third layer,provide a method for determining a global course of action for an agent that is used as an initialization to the differential equations from thefirst layer,guiding the agent and keeping it from getting stuck in local minima.

In thefirst layer,through the use of nonlinear dynamical sys-tems theory,we characterize in a mathematically precise way the behavior of our agents in complex dynamic environments.The agents lie in a constantly changing environment consisting of ob-stacles,targets and other agents.Depending on the application, agents reach one or multiple goals,while avoiding multiple ob-stacles.Both targets and obstacles can be static and/or moving.

Our agent modeling is based on the coupling of a set of nonlin-ear dynamical systems.Thefirst dynamical system is responsible for the control of the agent’s angular velocity.It uses carefully de-signed nonlinear attractor and repeller functions for targets and obstacles,respectively,to change the facing direction.Due to the nonlinearity of these functions a direct summation can generate undesired attractors that would lead to collisions and other unsuit-able behaviors.To remedy this problem we use a second nonlin-ear dynamical system which automatically computes the correct weighted contribution of the above functions at each time instant.

A third dynamical system controls the agent’s forward velocity which,depending on the situation may or may not take into ac-count the value of the angular velocity.

To model low-level personality attributes,such as agility or ag-gressiveness,we extend the above set of equations through ad-ditional parameterization.The attributes are modeled using this additional parameterization of the governing equations.Agents with different personalities will react differently in the same en-vironment,given the same set of initial conditions.Our approach is general,and other low-level behaviors can be modeled easily if needed in a given application.

Each agent is described by its position,geometrical shape, heading angle,forward velocity and personality.The nonlinear dynamical systems which model the changes in the above param-eters are based on local decisions.Therefore,to avoid local min-ima in the solution and to generate a nominal global trajectory we use harmonic function theory.This nominal trajectory is com-puted using only the static objects in the environment.Harmonic functions are solutions to the Laplace equation,and they create